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DanielVarga comments on Explained: Gödel's theorem and the Banach-Tarski Paradox - Less Wrong Discussion

10 Post author: XiXiDu 06 January 2012 05:23PM

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Comment author: DanielVarga 07 January 2012 01:32:53PM 2 points [-]

I seem to recall reading about a way to divide the interval [0, 1] in subsets, translating some of them, and getting [0, 2] (involving the Vitali set or something like that), but maybe my memory fails me.

This is possible if you use infinitely many subsets. With an uncountably infinite number of pieces it is true by definition, with a countably infinite number of pieces it can be proven using the Vitali set, and with a finite number of pieces it is not true.

Or am I missing something?

What Oscar_Cunningham said, but basically, no, you are not.

Comment author: [deleted] 07 January 2012 02:10:46PM 0 points [-]

Or am I missing something?

What Oscar_Cunningham said, but basically, no, you are not.

I was expecting something harder given that you called it "a nice exercise", so I pretty much assumed that mine was not the right solution...

Comment author: DanielVarga 08 January 2012 12:55:39AM 0 points [-]

Okay. In the original formulation of the paradox, the task is to cut a ball into pieces, and assemble two balls from the pieces. If I am not mistaken, you have solved a slightly easier task: cut a ball into pieces, and covered two balls with the pieces (with overlaps). A part of the "nice exercise" is to bridge this gap.

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